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Why can you turn clothing right-side-out?

145 points| ivoflipse | 14 years ago |math.stackexchange.com | reply

33 comments

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[+] ctdonath|14 years ago|reply
It's a matter of people imputing notions of "special" where they don't belong. Hence the importance of "proof" in math.

ETA: People are surprised that they can turn a sweater inside-out thru a sleeve or neck hole ONLY because they've imputed a "special" ability to the largest hole in the garment. The mathematical concept of "proof" strips away such imputations, leaving surprisingly unsurprising results - in this case, you can reverse a garment by pulling it thru one of its holes, be it the largest, smallest, or even a tear, because they're all just holes with nothing inherently topologically special about them. In a larger social concept: people tend to impute special attributes to various things where such attribution is not warranted; people who understand the concept of mathematical proof are less likely to get caught up in such incorrect imputation.

[+] Aloisius|14 years ago|reply
Who are these people who are surprised that you can turn a sweater inside-out through other holes?

To verify, I just went around my office and asked a couple people if they thought you could. Every person answered yes.

[+] raverbashing|14 years ago|reply
Actually, in real clothes it would be difficult to turn it inside out through a small tear

Topologically possible, but with a physical constraint (or more likely, you increase the tear trying to do it)

[+] xyzzyz|14 years ago|reply
I didn't quite get your comment (I'm a non-native speaker). Could you please be more elaborate?
[+] olavk|14 years ago|reply
You cant turn a pair of trousers inside out through the belt loops though.
[+] acheron|14 years ago|reply
Love this question and all the answers. I find topology fascinating even though I understand maybe 5% of it. I took a decent amount of math as part of my CS degree, but beyond basic calculus it was concentrated in probability, stats, and linear algebra; never came near topology. In hindsight I wish I had taken more math, but as a 19-20 year old student at the time, I was happy to be done with it.

If I invented a time machine, sometimes I think my second use of it would be to give my college self class choice and scheduling advice.

[+] jacobolus|14 years ago|reply
Topology is fun, and not especially mysterious if you give it the time. One of those subjects that is at the same time highly abstract and exercises the visual–spatial–kinetic thinking part of your brain. You can learn it whenever you like! (For a motivated student, I think self study of mathematics topics is better than a course with fixed problems anyhow, because you can move at your own pace, and take any path you prefer. Requires some focus though.) I don’t have enough experience with all the various textbooks to compare them, but I thought Munkres was alright.
[+] mturmon|14 years ago|reply
Nice observation, and 100% agree. Here's how one of the most painstaking explicators put it in the thread:

"I'm summarizing about 200 years of mathematics, almost none of which is standardly taught to undergraduates at almost any university."

which made me feel a little better.

[+] pfortuny|14 years ago|reply
The real problem is: what would the world look like if the result were different?

That looks more interesting to me.

[+] karamazov|14 years ago|reply
For the result to be different, the real world would need to have a different sort of topology. For example, if we lived in a non-Hausdorff space, the result would potentially be different. A Hausdorff space is one where for any two different points, you can take a small ball around each point, and the two balls won't intersect. A non-Hausdorff space is just a space where this isn't true for at least one pair of points.

So, for example, if wormholes are real, the space we live in might not be Hausdorff. And if you had a wormhole sewn into your shirt in the right way, you wouldn't be able to turn it inside out.

[+] adavies42|14 years ago|reply
thank you! i love asking that question of math results.
[+] ars|14 years ago|reply
When I think of topology the first I do is remember that size means nothing in topology.

So the long sleeve of the shirt? Shrink it right down - you are left with just a hole, and no sleeve.

Then flatten out the curvature of the neck and other parts, and you are left with a flat sheet of clothing, with two holes in it. There is no inside or outside to this, meaning the two sides are interchangeable, and that's why you can turn the real shirt inside out.